Data for "A direct algebraic proof for the non-positivity of Liouvillian spectral values and controlling the dissipative gap of systems with normal Lindblad operators"

This file contains the numerical data presented in the paper "A direct algebraic proof for the non-positivity of Liouvillian spectral values and controlling the dissipative gap of systems with normal Lindblad operators" (arXiv:2504.02256). In particular, this concerns the finite-size scaling of the Liouvillian spectra for the infinite-range transverse-field Ising model in Fig. 2. We use field strength g=1, spontaneous emission rate gamma=1, and the coupling J=J_c(kappa) with kappa=1/2 such that the full Liouvillian L=L_0+D_\kappa is critical in the thermodynamic limit N->infinity, and the unperturbed Liouvillian L_0 in the ferromagnetic phase. Depending on the system size N, we used different quasi-exact numerical methods: full 4^N Liouville space diagonalization for 3<=N<=7, full block diagonalization in the symmetric subspace for 6<=N<=24, and Krylov subspace iterations for the symmetric subspace, targeting the ten eigenvalues with largest real parts for N>=10. (Some of the L_0 data at the largest system sizes N>=120 is not converged.) These correspond to the three folders in data.zip. The specral data is given in the form of text tables with one file for each system size.